it follows that the \(p\) -th Hecke eigenvalue of \(\mathrm {Sh}_t(f)\)  is \(\lambda _p\) , where \(T(p^2)\)  is the Hecke operator on \(S_{k+1/2}(N,\varepsilon )\)  and \(T(p)\)  is the Hecke operator
on \(S_{2k}(N/2,\varepsilon ^2)\) . By [1]} we have
\(\sum _{\nu \ge 0}a(tp^{2\nu })X^\nu =a(t)\dfrac{1-\varepsilon _{N,t}(p)p^{k-1}X}{1-\lambda _pX+\varepsilon (p)^2p^{2k-1}X^2},\) 
